You don't really describe the circumstances as to what occurred. Was you dog on a leash? Did the other dog attack your dog? I assume your dog needed to be taken to the veterinarian as a result of the bite? Do you have any way of identifying the person whose dog bit your dog? If so, you might be able to pursue a small claims action against him or her for the costs you incurred, depending on the circumstances and assuming the other dog was the aggressor. Of course, if you can't find the dog owner you may simply be out of luck...
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Unless the dog is a great deal bigger than yours I think you should just pay your bill. At the doggy park people often let their dogs run loose and that is a risk we take.
My name is Stephen R. Cohen and have practiced since 1974. I practice in Los Angeles and Orange County, CA. These answers do not create an attorney client relationship. My answers may offend I believe in telling the truth, I use common sense as well as the law. Other state's laws may differ.. There are a lot of really good attorneys on this site, I will do limited appearances which are preparation of court documents it is , less expensive. However generally I believe an attorney is better than none, but many will offer a free consultation and a face to face meeting generally will be better, I like my clients to write a short one page history of the fact and questions they have prior to meeting with them, so nothing is forgotten.